The Chromatic Symmetric Function of a Graph Centred at a Vertex - Aliniaeifard, Farid; Wang, Victor; van Willigenburg, Stephanie (The Electronic Journal of Combinatorics, 2024)
We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of e-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function Y_G in noncommuting variables in this quotient algebra, which defines y_G : v, the chromatic symmetric function of a graph G centred at a vertex v. We then apply our methods to y_G : v and find new families of unit interval graphs that are (e)-positive, a stronger condition than classical e-positivity, thus confirming new cases of the (3+1)-free conjecture of Stanley and Stembridge. In our study of y_G : v, we also describe methods of constructing new e-positive graphs from given (e)-positive graphs and classify the (e)-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of y_G : v to acyclic orientations of graphs, including a noncommutative refinement of Stanley’s sink theorem.